We develop a physics-informed neural network (PINN) framework for parameter estimation in fractional-order SEIRD epidemic models. By embedding the Caputo fractional derivative into the network residuals via the L1 discretization scheme, our method simultaneously reconstructs epidemic trajectories and infers both epidemiological parameters and the fractional memory order $α$. The fractional formulation extends classical integer-order models by capturing long-range memory effects in disease progression, incubation, and recovery. Our framework learns the fractional memory order $α$ as a trainable parameter while simultaneously estimating the epidemiological rates $(β, σ, γ, μ)$. A composite loss combining data misfit, physics residuals, and initial conditions, with constraints on positivity and population conservation, ensures both accuracy and biological consistency. Tests on synthetic Mpox data confirm reliable recovery of $α$ and parameters under noise, while applications to COVID-19 show that optimal $α\in (0, 1]$ captures memory effects and improves predictive performance over the classical SEIRD model. This work establishes PINNs as a robust tool for learning memory effects in epidemic dynamics, with implications for forecasting, control strategies, and the analysis of non-Markovian epidemic processes.

In this work, we introduce a new class of neural network operators designed to handle problems where memory effects and randomness play a central role. In this work, we introduce a new class of neural network operators designed to handle problems where memory effects and randomness play a central role. These operators merge symmetrized activation functions, Caputo-type fractional derivatives, and stochastic perturbations introduced via Itô type noise. The result is a powerful framework capable of approximating functions that evolve over time with both long-term memory and uncertain dynamics. We develop the mathematical foundations of these operators, proving three key theorems of Voronovskaya type. These results describe the asymptotic behavior of the operators, their convergence in the mean-square sense, and their consistency under fractional regularity assumptions. All estimates explicitly account for the influence of the memory parameter $α$ and the noise level $σ$. As a practical application, we apply the proposed theory to the fractional Navier-Stokes equations with stochastic forcing, a model often used to describe turbulence in fluid flows with memory. Our approach provides theoretical guarantees for the approximation quality and suggests that these neural operators can serve as effective tools in the analysis and simulation of complex systems. By blending ideas from neural networks, fractional calculus, and stochastic analysis, this research opens new perspectives for modeling turbulent phenomena and other multiscale processes where memory and randomness are fundamental. The results lay the groundwork for hybrid learning-based methods with strong analytical backing.

Neural differential equation models have garnered significant attention in recent years for their effectiveness in machine learning applications.Among these, fractional differential equations (FDEs) have emerged as a promising tool due to their ability to capture memory-dependent dynamics, which are often challenging to model with traditional integer-order approaches.While existing models have primarily focused on constant-order fractional derivatives, variable-order fractional operators offer a more flexible and expressive framework for modeling complex memory patterns. In this work, we introduce the Neural Variable-Order Fractional Differential Equation network (NvoFDE), a novel neural network framework that integrates variable-order fractional derivatives with learnable neural networks.Our framework allows for the modeling of adaptive derivative orders dependent on hidden features, capturing more complex feature-updating dynamics and providing enhanced flexibility. We conduct extensive experiments across multiple graph datasets to validate the effectiveness of our approach.Our results demonstrate that NvoFDE outperforms traditional constant-order fractional and integer models across a range of tasks, showcasing its superior adaptability and performance.

Designing re-entry vehicles requires accurate predictions of hypersonic flow around their geometry. Rapid prediction of such flows can revolutionize vehicle design, particularly for morphing geometries. We evaluate advanced neural operator models such as Deep Operator Networks (DeepONet), parameter-conditioned U-Net, Fourier Neural Operator (FNO), and MeshGraphNet, with the objective of addressing the challenge of learning geometry-dependent hypersonic flow fields with limited data. Specifically, we compare the performance of these models for two grid types: uniform Cartesian and irregular grids. To train these models, we use 36 unique elliptic geometries for generating high-fidelity simulations with a high-order entropy-stable DGSEM solver, emphasizing the challenge of working with a scarce dataset. We evaluate and compare the four operator-based models for their efficacy in predicting hypersonic flow field around the elliptic body. Moreover, we develop a novel framework, called Fusion DeepONet, which leverages neural field concepts and generalizes effectively across varying geometries. Despite the scarcity of training data, Fusion DeepONet achieves performance comparable to parameter-conditioned U-Net on uniform grids while it outperforms MeshGraphNet and vanilla DeepONet on irregular, arbitrary grids. Fusion DeepONet requires significantly fewer trainable parameters as compared to U-Net, MeshGraphNet, and FNO, making it computationally efficient. We also analyze the basis functions of the Fusion DeepONet model using Singular Value Decomposition. This analysis reveals that Fusion DeepONet generalizes effectively to unseen solutions and adapts to varying geometries and grid points, demonstrating its robustness in scenarios with limited training data.

Many organisms and cell types, from bacteria to cancer cells, exhibit a remarkable ability to adapt to fluctuating environments. Additionally, cells can leverage memory of past environments to better survive previously-encountered stressors. From a control perspective, this adaptability poses significant challenges in driving cell populations toward extinction, and is thus an open question with great clinical significance. In this work, we focus on drug dosing in cell populations exhibiting phenotypic plasticity. For specific dynamical models switching between resistant and susceptible states, exact solutions are known. However, when the underlying system parameters are unknown, and for complex memory-based systems, obtaining the optimal solution is currently intractable. To address this challenge, we apply reinforcement learning (RL) to identify informed dosing strategies to control cell populations evolving under novel non-Markovian dynamics. We find that model-free deep RL is able to recover exact solutions and control cell populations even in the presence of long-range temporal dynamics.

This paper introduces a novel methodology for solving distributed-order fractional differential equations using a physics-informed machine learning framework. The core of this approach involves extending the support vector regression (SVR) algorithm to approximate the unknown solutions of the governing equations during the training phase. By embedding the distributed-order functional equation into the SVR framework, we incorporate physical laws directly into the learning process. To further enhance computational efficiency, Gegenbauer orthogonal polynomials are employed as the kernel function, capitalizing on their fractional differentiation properties to streamline the problem formulation. Finally, the resulting optimization problem of SVR is addressed either as a quadratic programming problem or as a positive definite system in its dual form. The effectiveness of the proposed approach is validated through a series of numerical experiments on Caputo-based distributed-order fractional differential equations, encompassing both ordinary and partial derivatives.

To effectively predict and understand these complex systems, mathematical models are employed, allowing to gain insights into the system behaviour without the need for time-consuming or expensive experiments. Due to the inherent presence of continuous dynamics in these systems, Differential Equations (DEs) are commonly employed as mathematical models, accounting for the continuous evolution of the system's behaviour and offering the advantage of enabling predictions throughout the entire time domain and not only at specific points. With the emergence of Neural Networks (NNs) and their impressive performance in fitting mathematical models to data, numerous studies have focused on modelling realworld systems. However, conventional NNs are designed to model discrete functions and may not be able to accurately capture the continuous dynamics observed in several systems. To overcome this limitation, Chen et al. [1] introduced the Neural Ordinary Differential Equations (Neural ODEs), a NN architecture that adjusts an Ordinary Differential Equation (ODE) to the dynamics of a system.

This paper presents a novel operational matrix method to accelerate the training of fractional Physics-Informed Neural Networks (fPINNs). Our approach involves a non-uniform discretization of the fractional Caputo operator, facilitating swift computation of fractional derivatives within Caputo-type fractional differential problems with $0<\alpha<1$. In this methodology, the operational matrix is precomputed, and during the training phase, automatic differentiation is replaced with a matrix-vector product. While our methodology is compatible with any network, we particularly highlight its successful implementation in PINNs, emphasizing the enhanced accuracy achieved when utilizing the Legendre Neural Block (LNB) architecture. LNB incorporates Legendre polynomials into the PINN structure, providing a significant boost in accuracy. The effectiveness of our proposed method is validated across diverse differential equations, including Delay Differential Equations (DDEs) and Systems of Differential Algebraic Equations (DAEs). To demonstrate its versatility, we extend the application of the method to systems of differential equations, specifically addressing nonlinear Pantograph fractional-order DDEs/DAEs. The results are supported by a comprehensive analysis of numerical outcomes.

The primary goal of this research is to propose a novel architecture for a deep neural network that can solve fractional differential equations accurately. A Gaussian integration rule and a $L_1$ discretization technique are used in the proposed design. In each equation, a deep neural network is used to approximate the unknown function. Three forms of fractional differential equations have been examined to highlight the method's versatility: a fractional ordinary differential equation, a fractional order integrodifferential equation, and a fractional order partial differential equation. The results show that the proposed architecture solves different forms of fractional differential equations with excellent precision.